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Theorem imaiun1 43640
Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
Assertion
Ref Expression
imaiun1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem imaiun1
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3285 . . . 4 (∃𝑥𝐴𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑧𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
2 vex 3481 . . . . . 6 𝑦 ∈ V
32elima3 6086 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
43rexbii 3091 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
5 eliun 4999 . . . . . . 7 (⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵)
65anbi2i 623 . . . . . 6 ((𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ (𝑧𝐶 ∧ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵))
7 r19.42v 3188 . . . . . 6 (∃𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑧𝐶 ∧ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵))
86, 7bitr4i 278 . . . . 5 ((𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
98exbii 1844 . . . 4 (∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑧𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
101, 4, 93bitr4ri 304 . . 3 (∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
112elima3 6086 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵))
12 eliun 4999 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
1310, 11, 123bitr4i 303 . 2 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ 𝑦 𝑥𝐴 (𝐵𝐶))
1413eqriv 2731 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1536  wex 1775  wcel 2105  wrex 3067  cop 4636   ciun 4995  cima 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-iun 4997  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701
This theorem is referenced by:  trclimalb2  43715
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